User blog:Nayuta Ito/Googol series from the new source
Many googol series articles have the link to this page, but actually, it's outdated. So, I'm going to summarize the new "improved" page. Broken Notation The definition of Ackermann's Generalized Exponential Notation, which is used throughout the page, says: g(0, a, b) = b + a (addition) g(1, a, b) = ab = a*b = a++b (multiplication) g(2, a, b) = ba = a^b = a↑b = a**b = a+++b (exponentiation) g(3, a, b} = ab = a^^b = a↑↑b (tetration) g(a, 0, 1) = 1 (zeroth power) g(c, 0, a) = 1 (zeroth operation) g(a, b, c) = g(a - 1, g(a -1 , b, c) , c) (expansion) g(a, b, c, d) = g(a - 1, b, c, g(b, c, d)) (nesting about base number) g(a, b, c, d, e) = g(a - 1, b, g(c, d, e), e) (nesting about power number) g(a, b, c, d., e, f) = g(a - 1, b, c, g(d, e, f), e, f) (nesting about operation number) (I just copy-and-pasted, so the typos are original. The page has so many typos!) There are some problems in the definition. First, there is no base rule. If the g function has four or more operands, there is no way to terminate the calculation! Also, the rule for g(a,b,c) is not consistent with the first four operations. (Maybe he thought the arrow notation was left-associative, which makes the definition consistent) For example: g(3,3,3)=3^^3, according to the g(3,a,b) rule. g(3,3,3)=g(2,g(2,3,3),3)=(3^3)^3, according to the g(a,b,c) rule and g(2,a,b) rule. Note that g(a,b,c) is notactually expansion, neither. If we interpret that g(a,b,c) rule is applied for a>=4, then it's just as strong as hexation. Numbers The first number below the definition is already broken: n-greats googol g(2, g(2, g(2, 1, 1, 3/2, 2), 10), 10) = g(2, g(2, g(1, g(1, 3/2, 2), 2), 10), 10) = g(2, g(2, 5, 10), 10) There are two strange things. Even though the name looks like to have a varieble n, but it's actually supposed to be one number. More atrangely, the numbers from the two definitions (ignore the first five-argument-using one) are different! The middle one suggests 10106, but the right one suggests 10105. Also, the description by words suggests 10102*(3/2)n. The next number makes more sense: n-greats googolplex = g(2, n + 2, g(2, 100, g(3, 2, 10))). The formula indicates 101000(n+2). Anything works right this time, except that the writer believed the exponentiation was communitive. (Skipped a bit because it's not broken except for the five-argument g) The next part is almost completely valid. It's all about googo- prefix, with some valid examples. But I found a mistake for it: quadrix = four-nines = 929 = 9999 = 4á9ñ = g(2, 5, 10) - 1 googolquadrix = g(2, g(1. 2, 9999), 9999) = g( 2, 19998, 9999) > g(2, 43000, 10) Since googo-n is supposed to be 2n^n, so it should be g(2,9999,g(1,2,9999). Also, the letter L in "googolquadrix" should not be. Here's funnier definition: bigoogol = g(1, g(1, g(1, g(2, log(2, 50, 100), 10)), g(2, 50, 100)) = g(1, g(2, 201, 10), g(2, 100, 10) Did you notice that there is a THREE-VARIABLE LOG in the definition? It's not just funny but also brackets are unmatched! Although the right side formula can be interpreted as 10301, which does not match with the Bigoogol page in Googology Wiki. The next part is about , which makes perfect sense except for some typos. The next part is googolbar and some relatives, Except for the "barbaric" part, everything works. The next part is gag-, megafuga-, and some extended SI prefixes. Googol series The most important part in the page. First, the "oo" in "googol" is extended. This is how is goes: *(gogol=50^50 implied) *googol=100^50 *geegol=150^50 *gorgol=200^50 *giegol=250^50 *ghigol=300^50 *gegol=350^50 *geigol=400^50 *gegol=450^50 *gengol=500^50 *gelgol=550^50 (originally it was a typo for 600^50) *gwelgol=600^50 *girgol=650^50 And next, the more generalized formula g-n-g-m-k=nkmk is indicated. The next line does not make much sense, but I guess it indicates: g-n-gt-m-k = (n\uparrow^t k) \uparrow^{k+1} (m \uparrow k). Below nonsence more-than-four-arguments part continues to the end. Summary THe author themself does not understand the g function perfectly. They always mistake g(1,a,b) as a+b. We must force them to update the page! Category:Blog posts